🌐Lie Groups

Lie Gropus

c++ type	Lie group name	Description	minimal representation	#DoF	Matrix representation	compact internal manifold representation	#params
Rotation2<T>	Special Orthogonal Group in 2D, SO(2)	rotations in 2d, also called Circle Group, or just "angle"	rotation angle	1	2x2 matrix	unit complex number	2
Rotation3<T>	Special Orthogonal Group in 3D, SO(3)	rotations in 3d, 3D orientations	rotation vector	3	3x3 matrix	unit quaternion number	4
Isometry2<T>	Special Euclidean Group in 2D, SE(3)	rotations and translations in 2D, also called 2D rigid body transformations, 2d poses, plane isometries	tangent vector of SE(2)	3	3x3 matrix	unit complex number + translation vector	2+2 = 4
Isometry3<T>	Special Euclidean Group in 3D, SE(3)	rotations and translations in 3D, also called rigid body transformations,6 DoF poses, Euclidean isometries	tangent vector of R x SE(3)	6	4x4 matrix	unit quaternion number + translation vector	4+3 = 7
RxSo2<T>	Direct product of SO(3) and scalar matrix, R x SO(2)	scaling and rotations in 2D	tangent vector of R x SO(2)	3	2x2 matrix	non-zero complex number	2
RxSo3<T>	Direct product of SO(3) and scalar matrix R x SO(3)	scaling and rotations in 3D	tangent vector of R x SO(3)	4	3x3 matrix	non-zero quaternion number	4
Similarity2<T>	Similarity Group in 2D, Sim(2)	scaling, rotations and translation in 2D	tangent vector of Sim(2)	4	3x3 matrix	non-zero complex number+ translation vector	2+2 = 4
Similarity3<T>	Similarity Group in 3D, Sim(3)	scaling, rotations and translation in 3D	tangent vector of Sim(3)	4	4x4 matrix	non-zero complex number+ translation vector	4+3 = 7
Cartesian2<T>	2D Euclidean Vector Space, R^2	all vector spaces are trivial Lie groups, also called 2d translation group, the translation part of SE(2)	2-vector	2	3x3 matrix	2-vector	2
Cartesian3<T>	3D Euclidean Vector Space, R^3	all vector spaces are trivial Lie groups, also called 3d translation group, the translation part of SE(2)	3-vector	3	4x4 matrix	3-vector	3